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Question 22922: All Distinct Point Pairs Must Be Separated in the Complete Finite Topology

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0044-22922 · q-20261014-advance-20261111-r03-hausdorff-space-0047

Original question and givens

Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

  • G001 — Use finite unique labeled points=[a21, b21, c21] and the complete canonical topology=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Give (points, pair_witness_rows, Hausdorff, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source operation=hausdorff-full-audit;original snapshot and mathematics retained

After

explicit ordered source output fields=[points, pair_witness_rows, Hausdorff, certificate]

Conditions: none

Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []];actual open index=0,U=[a21, b21, c21]

After

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [a21, b21, c21]], [1, [a21, b21, c21], [a21, b21]], [2, [a21, b21, c21], [a21, c21]], [3, [a21, b21, c21], [a21]], [4, [a21, b21, c21], [b21, c21]], [5, [a21, b21, c21], [b21]], [6, [a21, b21, c21], [c21]], [7, [a21, b21, c21], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S001

S003 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []];actual open index=1,U=[a21, b21]

After

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [a21, b21]], [1, [a21, b21], [a21, b21]], [2, [a21, b21, c21], [a21]], [3, [a21, b21], [a21]], [4, [a21, b21, c21], [b21]], [5, [a21, b21], [b21]], [6, [a21, b21, c21], []], [7, [a21, b21], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S002

S004 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []];actual open index=2,U=[a21, c21]

After

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [a21, c21]], [1, [a21, b21, c21], [a21]], [2, [a21, c21], [a21, c21]], [3, [a21, c21], [a21]], [4, [a21, b21, c21], [c21]], [5, [a21, b21, c21], []], [6, [a21, c21], [c21]], [7, [a21, c21], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S003

S005 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []];actual open index=3,U=[a21]

After

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [a21]], [1, [a21, b21], [a21]], [2, [a21, c21], [a21]], [3, [a21], [a21]], [4, [a21, b21, c21], []], [5, [a21, b21], []], [6, [a21, c21], []], [7, [a21], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S004

S006 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []];actual open index=4,U=[b21, c21]

After

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [b21, c21]], [1, [a21, b21, c21], [b21]], [2, [a21, b21, c21], [c21]], [3, [a21, b21, c21], []], [4, [b21, c21], [b21, c21]], [5, [b21, c21], [b21]], [6, [b21, c21], [c21]], [7, [b21, c21], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S005

S007 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []];actual open index=5,U=[b21]

After

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [b21]], [1, [a21, b21], [b21]], [2, [a21, b21, c21], []], [3, [a21, b21], []], [4, [b21, c21], [b21]], [5, [b21], [b21]], [6, [b21, c21], []], [7, [b21], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S006

S008 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []];actual open index=6,U=[c21]

After

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], [c21]], [1, [a21, b21, c21], []], [2, [a21, c21], [c21]], [3, [a21, c21], []], [4, [b21, c21], [c21]], [5, [b21, c21], []], [6, [c21], [c21]], [7, [c21], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S007

S009 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];source canonical open family=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []];actual open index=7,U=[]

After

ALL [other_index,union,intersection] rows=[[0, [a21, b21, c21], []], [1, [a21, b21], []], [2, [a21, c21], []], [3, [a21], []], [4, [b21, c21], []], [5, [b21], []], [6, [c21], []], [7, [], []]];empty/full present;family qualification passed with exact source English collation

Conditions: none

Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].

Step identifiers and dependencies

Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S008

S010 · implies

Before

Hausdorff ORIGINAL pair index0:ORIGINAL distinct carrier points x=a21,y=b21;source open order=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a21, b21, c21], No], [0, 1, [a21, b21], No], [0, 4, [b21, c21], No], [0, 5, [b21], No], [1, 0, [a21, b21], No], [1, 1, [a21, b21], No], [1, 4, [b21], No], [1, 5, [b21], No], [2, 0, [a21, c21], No], [2, 1, [a21], No], [2, 4, [c21], No], [2, 5, [], Yes]];FIRST source-order witness=[[a21, c21], [b21]]

Conditions: none

Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample.

Step identifiers and dependencies

Rule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S009

S011 · implies

Before

Hausdorff ORIGINAL pair index1:ORIGINAL distinct carrier points x=a21,y=c21;source open order=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a21, b21, c21], No], [0, 2, [a21, c21], No], [0, 4, [b21, c21], No], [0, 6, [c21], No], [1, 0, [a21, b21], No], [1, 2, [a21], No], [1, 4, [b21], No], [1, 6, [], Yes]];FIRST source-order witness=[[a21, b21], [c21]]

Conditions: none

Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample.

Step identifiers and dependencies

Rule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S010

S012 · implies

Before

Hausdorff ORIGINAL pair index2:ORIGINAL distinct carrier points x=b21,y=c21;source open order=[[a21, b21, c21], [a21, b21], [a21, c21], [a21], [b21, c21], [b21], [c21], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a21, b21, c21], No], [0, 2, [a21, c21], No], [0, 4, [b21, c21], No], [0, 6, [c21], No], [1, 0, [a21, b21], No], [1, 2, [a21], No], [1, 4, [b21], No], [1, 6, [], Yes]];FIRST source-order witness=[[a21, b21], [c21]]

Conditions: none

Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample.

Step identifiers and dependencies

Rule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S011

S013 · implies

Before

ORIGINAL carrier order=[a21, b21, c21];ALL unordered distinct pairs with first index<second=[[a21, b21], [a21, c21], [b21, c21]]

After

ALL pair-witness rows=[[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]];Hausdorff=Yes;no pairs means vacuous True,not failure

Conditions: none

One exhaustively failed pair refutes Hausdorffness;all pairs must succeed for this finite certificate.For a finite Hausdorff space,intersect x-neighborhoods excluding each other y to obtain open singleton{x},hence discreteness.Do not extrapolate that corollary to infinite spaces.

Step identifiers and dependencies

Rule: hausdorff_ALL_original_point_pairs_empty_singleton_vacuity · Uses: G001, S012

Reverse checks

  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([a21, b21, c21], [[a21, b21, [[a21, c21], [b21]]], [a21, c21, [[a21, b21], [c21]]], [b21, c21, [[a21, b21], [c21]]]], Yes, all_distinct_pairs_exhaustively_separated)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 68e92af8d23f1070bdde8720b8ff5966a5352a7f4eb87ffd3de7683cc8304f25
  • candidate record SHA-256: 2e92b39c2b425990010b90f32faad2d48af3074016a1a513eb995a207fb61078
  • semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
  • question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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